D-modules and hyperplane arrangements

Title and Abstract:

Dan Bath

Title: Hyperplane Arrangements Satisfy (un)Twisted Logarithmic Comparison Theorems.

Abstract: In the 1970s, Terao conjectured that hyperplane arrangements satisfy the logarithmic comparison theorem. That is, the cohomology of the logarithmic de Rham complex computes the cohomology of the complement of the arrangement with constant coefficients. One can also ask about a twisted version where the differential of the logarithmic complex is twisted by a one-form and the cohomology of the complement of the arrangement is given by the corresponding rank one local system. We will prove hyperplane arrangements satisfy the twisted logarithmic comparison theorem, subject to mild restrictions on the weights defining the twist. This implies the untwisted version, resolving Terao's aforementioned conjecture.

Unlike the Brieskorn algebra (i.e. the Orlik--Solomon algebra), which can only compute rank one local systems corresponding to non-torsion translated local systems, our twisted logarithmic comparison theorem computes all rank one local systems and, moreover, reduces to taking the cohomology of a complex of finite dimensional vector spaces.

We will discuss two proofs of this: the first, a purely commutative algebra proof by the speaker; the second, a D-module theoretic proof by the speaker and Saito. Applications will also be sketched.

Graham Denham
Title: Varchenko-Gelfand filtrations revisited

Abstract: I will recall how real hyperplane arrangements and, more generally, oriented matroids admit combinatorial filtrations on their face algebras.  Applications to random walks on the face algebra, homology classes in the Salvetti complex, and equivariant cohomology will be considered.

Hiroyuki Ochiai
Title: A non-holonomic D-module with infinite-dimensional solutions

Abstract: T.Tauchi (2018) introduced an interesting setting that an algebraic Lie group acting on a finite-dimensional real vector space with finitely many orbits admits inifite-dimensional invariant distributions.  This was a counter-example to a naive working hypotheis in branching problems of representation theory of Lie groups.  

In this talk, I will not come into groups and representations, and we would like to understand this example and this phenomea in terms of D-modules.

Uli Walther
Title:  Remarks on tautological systems
Abstract:  Joint with Paul Goerlach, Thomas Reichelt, Christian Sevenheck, Avi Steiner 

Let G be a complex Lie group, acting equivariantly on a very ample line bundle L of a projective variety X. Let G' be the product of G and the multiplicative group. Then G' acts on the space V of sections of L. Let D be the ring of differential operators on V and choose a character for G'. Then there is an induced D-module whose construction resembles that of the classical GKZ-systems. It was introduced by Hotta, re-introduced 10 years ago by Lian and Yau, and investigated by Bloch, Bong, Huang, Lian, Song, Srinivas, Yau in a series of papers; the motivation comes from mirror symmetry (which will not be discussed here).

We will talk about the definition of this module, some basic properties, and specifically discuss the case where X is a homogeneous G-space.